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A new spectral analysis of stationary random Schrödinger operators

Mitia Duerinckx, Christopher Shirley

Abstract

Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states, but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.

A new spectral analysis of stationary random Schrödinger operators

Abstract

Motivated by the long-time transport properties of quantum waves in weakly disordered media, the present work puts random Schrödinger operators into a new spectral perspective. Based on a stationary random version of a Floquet type fibration, we reduce the description of the quantum dynamics to a fibered family of abstract spectral perturbation problems on the underlying probability space. We state a natural resonance conjecture for these fibered operators: in contrast with periodic and quasiperiodic settings, this would entail that Bloch waves do not exist as extended states, but rather as resonant modes, and this would justify the expected exponential decay of time correlations. Although this resonance conjecture remains open, we develop new tools for spectral analysis on the probability space, and in particular we show how ideas from Malliavin calculus lead to rigorous Mourre type results: we obtain an approximate dynamical resonance result and the first spectral proof of the decay of time correlations on the kinetic timescale. This spectral approach suggests a whole new way of circumventing perturbative expansions and renormalization techniques.

Paper Structure

This paper contains 30 sections, 29 theorems, 299 equations.

Table of Contents

  1. Introduction
  2. General overview
  3. Summary of our approach and results
  4. Main results
  5. Framework
  6. Basic spectral theory of fibered operators
  7. Perturbative Mourre's commutator approach
  8. Exact resonance conjectures
  9. Stationary random setting and Floquet--Bloch fibration
  10. Stationary setting
  11. Stationary differential calculus
  12. Stationary Floquet transform
  13. Stationary Floquet--Bloch fibration
  14. Basic spectral theory of fibered operators
  15. Unperturbed fibered operators
  16. ...and 15 more sections

Key Result

Proposition 1

Given a stationary Gaussian field $V_0$ on $\mathbb R^d$ with covariance function $\mathcal{C}_0$, denote by $\widehat{\mathcal{C}}_0$ the (nonnegative measure) Fourier transform of $\mathcal{C}_0$, and assume that the probability space $(\Omega,\mathbb{P})$ is endowed with the $\sigma$-algebra $\si

Theorems & Definitions (57)

  • Proposition 1: Spectral decomposition of $H_{k,0}^{\operatorname{st}}$
  • Proposition 2: Spectrum of $H_{k,\lambda}^{\operatorname{st}}$
  • Proposition 3: Instability of the bound state
  • Theorem 4: Perturbative Mourre's theory up to truncation
  • Corollary 5: Exponential decay law on kinetic timescale
  • Remark 2.1: Continuous resonant spectrum
  • Corollary 6: Consequences of resonance conjectures
  • Proposition 7: Approximate computation of resonances
  • Remark 2.2: Full Rayleigh--Schršdinger series for resonances
  • Definition 3.1
  • ...and 47 more